Smoothing the domain out for positive solutions

Bulletin of the Australian Mathematical Society

Volumn 61, Issue 3, 2000, Pages 405-413

  Liu, Yaping ^a  

^a PITTSBURG STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034195642     PISSN: 00049727     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0004972700022437     Document Type: Article

References (27)

1. H. Amann, 'Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces', SIAM Rev. 18 (1976), 620-709.
2. J. Appell, 'The superposition operator in function spaces - a survey', Exposition Math. 6 (1988), 207-270.
3. I. Babuška and R. Výborný, 'Continuous dependence of eigenvalues on the domain', Czechoslovak Math. J. 15 (1965), 169-178.
4. H. Berestycki and P. L. Lions, 'Some applications of the method of super and subsolutions', in Bifurcation and Nonlinear Eigenvalue Problems, Lecture Notes in Mathematics 782 (Springer-Verlag, Berlin, Heidelberg, New York, 1980), pp. 16-41.
5. H. Berestycki, L. Nirenberg and S.R.S. Varadhan, 'The principal eigenvalue and maximum principle for second-order elliptic operators in general domains', Comm. Pure Appl. Math. 47 (1994), 47-92.
6. J. Blat and K.J. Brown, 'Bifurcation of steady-state solutions in predator-prey and competition systems', Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21-34.
7. K.J. Brown, 'Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations', J. Math. Anal. Appl. 95 (1983), 251-264.
8. E.N. Dancer, 'The effect of domain shape on the number of positive solutions of certain nonlinear equations', J. Differential Equations 74 (1988), 120-156.
9. E.N. Dancer, 'The effect of domain shape on the number of positive solutions of certain nonlinear equations, II', J. Differential Equations 87 (1990), 316-339.
10. E.N. Dancer and D. Daners, 'Domain perturbation for elliptic equations subject to Robin boundary conditions', J. Differential Equations 138 (1997), 86-132.
11. E.N. Dancer and K. Schmitt, 'On positive solutions of semilinear elliptic equations', Proc. Amer. Math. Soc. 101 (1987), 445-452.
12. D. Daners, 'Domain perturbation for linear and nonlinear parabolic equations', J. Differential Equations 129 (1996), 358-402.
13. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, (2nd edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1983).
14. S. Heikkilä, 'On an elliptic boundary value problem with discontinuous nonlinearity', Appl. Anal. 37 (1990), 183-189.
15. J. Hernández, 'Maximum principles and decoupling for positive solutions of reaction-diffusion systems', in Reaction-diffusion Equations, (K. J. Brown and A. A. Lacey, Editors), Oxford Science Publications, 1988 (Oxford University Press, New York, 1990), pp. 199-224.
16. E. Jahnke, F. Emde and F. Lösch, Tables of higher functions, (6th edition) (McGraw-Hill, New York, 1960).
17. J.R. Kuttler and V.G. Sigillito, 'Eigenvalues of the Laplacian in two dimensions', SIAM Rev. 26 (1984), 163-193.
18. A. Leung, Systems of nonlinear partial differential equations: Applications to biology and engineering, Mathematics and its Applications (Kluwer Academic Publishers, Dordrecht, 1989).
19. L. Li, 'Coexistence theorems of steady-states for predator-prey interacting systems', Trans. Amer. Math. Soc. 305 (1988), 143-166.
20. L. Li and A. Ghoreishi, 'On positive solutions of general nonlinear elliptic symbiotic interacting systems', Appl. Anal. 14 (1991), 281-295.
21. L. Li and Y. Liu, 'Spectral and nonlinear effects in certain elliptic systems of three variables', SIAM J. Math. Anal. 24 (1993), 480-498.
22. P.L. Lions, 'On the existence of positive solutions of semilinear elliptic equations', SIAM Rev. 24 (1982), 441-467.
23. J. Rauch and M. Taylor, 'Potential and scattering theory on wildly perturbed domains', J. Func. Anal. 18 (1975), 27-59.
24. I.V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Translations of Mathematical Monographs 139 (Amer. Math. Soc., Providence, RI, 1994).
25. F. Stummel, 'Perturbation theory for Sobolev spaces', Proc. Roy. Soc. Edinburgh, Sect. A 73 (1974), 5-49.
26. F. Stummel, Perturbation of domains in elliptic boundary value problems, Lecture Notes in Mathematics 503 (Springer-Verlag, Berlin, Heidelberg, New York, 1975), pp. 110-135.
27. G. Sweers, 'Semilinear elliptic problems on domains with corners', Comm. Partial Differential Equations 14 (1989), 1229-1247.